Assuming your data frame is named "x":
aggregate(x$B, list(cut(x$A, breaks=c(0, 200, 400))), mean)
# Group.1 x
# 1 (0,200] 90.0
# 2 (200,400] 107.5
With "data.table", you can do the following:
library(data.table)
as.data.table(x)[, .(RANGE = mean(B)), by = .(MEAN = cut(A, c(0, 200, 400)))]
# MEAN RANGE
# 1: (0,200] 90.0
# 2: (200,400] 107.5
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